Runtime Complexity (full) proof of /tmp/tmp8ZeqVN/PEANO_nosorts_noand

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).

0 CpxTRS

↳1 DecreasingLoopProof (⇔, 390 ms)

↳2 BOUNDS(n^1, INF)

↳3 RenamingProof (⇔, 0 ms)

↳4 CpxRelTRS

    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)

    ↳6 typed CpxTrs

    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)

    ↳8 typed CpxTrs

    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 253 ms)

    ↳10 typed CpxTrs

    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 9 ms)

    ↳12 typed CpxTrs

    ↳13 RewriteLemmaProof (LOWER BOUND(ID), 110 ms)

    ↳14 BEST

        ↳15 typed CpxTrs

            ↳16 RewriteLemmaProof (LOWER BOUND(ID), 268 ms)

            ↳17 BEST

                ↳18 typed CpxTrs

                    ↳19 RewriteLemmaProof (LOWER BOUND(ID), 183 ms)

                    ↳20 BEST

                        ↳21 typed CpxTrs

                            ↳22 RewriteLemmaProof (LOWER BOUND(ID), 175 ms)

                            ↳23 BEST

                                ↳24 typed CpxTrs

                                    ↳25 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)

                                    ↳26 BEST

                                        ↳27 typed CpxTrs

                                            ↳28 RewriteLemmaProof (LOWER BOUND(ID), 108 ms)

                                            ↳29 BEST

                                                ↳30 typed CpxTrs

                                                    ↳31 RewriteLemmaProof (LOWER BOUND(ID), 128 ms)

                                                    ↳32 BEST

                                                        ↳33 typed CpxTrs

                                                            ↳34 LowerBoundsProof (⇔, 0 ms)

                                                            ↳35 BOUNDS(n^2, INF)

                                                        ↳36 typed CpxTrs

                                                            ↳37 LowerBoundsProof (⇔, 0 ms)

                                                            ↳38 BOUNDS(n^2, INF)

                                                ↳39 typed CpxTrs

                                                    ↳40 LowerBoundsProof (⇔, 0 ms)

                                                    ↳41 BOUNDS(n^2, INF)

                                        ↳42 typed CpxTrs

                                            ↳43 LowerBoundsProof (⇔, 0 ms)

                                            ↳44 BOUNDS(n^2, INF)

                                ↳45 typed CpxTrs

                                    ↳46 LowerBoundsProof (⇔, 0 ms)

                                    ↳47 BOUNDS(n^2, INF)

                        ↳48 typed CpxTrs

                            ↳49 LowerBoundsProof (⇔, 0 ms)

                            ↳50 BOUNDS(n^2, INF)

                ↳51 typed CpxTrs

                    ↳52 LowerBoundsProof (⇔, 0 ms)

                    ↳53 BOUNDS(n^2, INF)

        ↳54 typed CpxTrs

            ↳55 LowerBoundsProof (⇔, 0 ms)

            ↳56 BOUNDS(n^1, INF)

(0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0) → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0) → 0
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Rewrite Strategy: FULL

(1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(n¹):
The rewrite sequence
a__U11(tt, M, U11(tt, X257_4, X358_4)) →⁺ s(a__plus(a__U11(tt, X257_4, X358_4), mark(M)))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].
The pumping substitution is [X358_4 / U11(tt, X257_4, X358_4)].
The result substitution is [M / X257_4].

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

(4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

S is empty.
Rewrite Strategy: FULL

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(6) Obligation:

TRS:
Rules:
a__U11(tt, M, N) → a__U12(tt, M, N)
a__U12(tt, M, N) → s(a__plus(mark(N), mark(M)))
a__plus(N, 0') → mark(N)
a__plus(N, s(M)) → a__U11(tt, M, N)
mark(U11(X1, X2, X3)) → a__U11(mark(X1), X2, X3)
mark(U12(X1, X2, X3)) → a__U12(mark(X1), X2, X3)
mark(plus(X1, X2)) → a__plus(mark(X1), mark(X2))
mark(tt) → tt
mark(s(X)) → s(mark(X))
mark(0') → 0'
a__U11(X1, X2, X3) → U11(X1, X2, X3)
a__U12(X1, X2, X3) → U12(X1, X2, X3)
a__plus(X1, X2) → plus(X1, X2)

Types:
a__U11 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
tt :: tt:s:0':U11:U12:plus
a__U12 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
s :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
a__plus :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
mark :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
0' :: tt:s:0':U11:U12:plus
U11 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
U12 :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
plus :: tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus → tt:s:0':U11:U12:plus
hole_tt:s:0':U11:U12:plus1_0 :: tt:s:0':U11:U12:plus
gen_tt:s:0':U11:U12:plus2_0 :: Nat → tt:s:0':U11:U12:plus

(7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:
a__U11, a__U12, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(8) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__U11, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__U12.

(10) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__plus.

(12) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus2_0(+(n7802_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0))) →_IH
s(gen_tt:s:0':U11:U12:plus2_0(c7803_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(14) Complex Obligation (BEST)

(15) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12, a__plus

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8498₀ + n8498₀ + n8498₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n8498_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n8498_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0))))) →_L^{Ω(2 + n8498₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n8498_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(17) Complex Obligation (BEST)

(18) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12, a__plus, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n10562₀ + n10562₀ + n10562₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n10562_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n10562_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n10562_0))))) →_L^{Ω(2 + n10562₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n10562_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(20) Complex Obligation (BEST)

(21) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8498₀ + n8498₀ + n8498₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n10562₀ + n10562₀ + n10562₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__plus, a__U11, mark

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Induction Base:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, 0)))

Induction Step:
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, +(n12888_0, 1)))) →_R^Ω(1)
a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)), gen_tt:s:0':U11:U12:plus2_0(a)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)), gen_tt:s:0':U11:U12:plus2_0(a)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(a)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))))) →_L^{Ω(1 + a)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))))) →_L^{Ω(2 + n12888₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0)))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(23) Complex Obligation (BEST)

(24) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
mark, a__U11, a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(25) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)

Induction Base:
mark(gen_tt:s:0':U11:U12:plus2_0(0)) →_R^Ω(1)
tt

Induction Step:
mark(gen_tt:s:0':U11:U12:plus2_0(+(n15871_0, 1))) →_R^Ω(1)
s(mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0))) →_IH
s(gen_tt:s:0':U11:U12:plus2_0(c15872_0))

We have rt ∈ Ω(n¹) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n).

(26) Complex Obligation (BEST)

(27) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8498₀ + n8498₀ + n8498₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n10562₀ + n10562₀ + n10562₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U11, a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(28) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)

Induction Base:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n16876_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(+(n16876_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0))))) →_L^{Ω(2 + n16876₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n16876_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(29) Complex Obligation (BEST)

(30) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n10562₀ + n10562₀ + n10562₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

The following defined symbols remain to be analysed:
a__U12

They will be analysed ascendingly in the following order:
a__U11 = a__U12
a__U11 = a__plus
a__U11 = mark
a__U12 = a__plus
a__U12 = mark
a__plus = mark

(31) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n20155₀ + n20155₀ + n20155₀²)

Induction Base:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(c))

Induction Step:
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(+(n20155_0, 1)), gen_tt:s:0':U11:U12:plus2_0(c)) →_R^Ω(1)
s(a__plus(mark(gen_tt:s:0':U11:U12:plus2_0(c)), mark(gen_tt:s:0':U11:U12:plus2_0(+(n20155_0, 1))))) →_L^{Ω(1 + c)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), mark(gen_tt:s:0':U11:U12:plus2_0(+(1, n20155_0))))) →_L^{Ω(2 + n20155₀)}
s(a__plus(gen_tt:s:0':U11:U12:plus2_0(c), gen_tt:s:0':U11:U12:plus2_0(+(1, n20155_0)))) →_R^Ω(1)
s(a__U11(tt, gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_R^Ω(1)
s(a__U12(tt, gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c))) →_IH
s(*3_0)

We have rt ∈ Ω(n²) and sz ∈ O(n). Thus, we have irc_R ∈ Ω(n²).

(32) Complex Obligation (BEST)

(33) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n20155₀ + n20155₀ + n20155₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(34) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)

(35) BOUNDS(n^2, INF)

(36) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n20155_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n20155₀ + n20155₀ + n20155₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(37) LowerBoundsProof (EQUIVALENT transformation)

(38) BOUNDS(n^2, INF)

(39) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n15871_0)) → gen_tt:s:0':U11:U12:plus2_0(n15871_0), rt ∈ Ω(1 + n15871₀)
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n16876_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n16876₀ + n16876₀ + n16876₀²)
a__U12(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n10562_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n10562₀ + n10562₀ + n10562₀²)
a__plus(gen_tt:s:0':U11:U12:plus2_0(a), gen_tt:s:0':U11:U12:plus2_0(+(1, n12888_0))) → *3_0, rt ∈ Ω(a·n12888₀ + n12888₀ + n12888₀²)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(40) LowerBoundsProof (EQUIVALENT transformation)

(41) BOUNDS(n^2, INF)

(42) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(43) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n²) was proven with the following lemma:
a__U11(gen_tt:s:0':U11:U12:plus2_0(0), gen_tt:s:0':U11:U12:plus2_0(n8498_0), gen_tt:s:0':U11:U12:plus2_0(c)) → *3_0, rt ∈ Ω(c·n8498₀ + n8498₀ + n8498₀²)

(44) BOUNDS(n^2, INF)

(45) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(46) LowerBoundsProof (EQUIVALENT transformation)

(47) BOUNDS(n^2, INF)

(48) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(49) LowerBoundsProof (EQUIVALENT transformation)

(50) BOUNDS(n^2, INF)

(51) Obligation:

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(52) LowerBoundsProof (EQUIVALENT transformation)

(53) BOUNDS(n^2, INF)

(54) Obligation:

Lemmas:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

Generator Equations:
gen_tt:s:0':U11:U12:plus2_0(0) ⇔ tt
gen_tt:s:0':U11:U12:plus2_0(+(x, 1)) ⇔ s(gen_tt:s:0':U11:U12:plus2_0(x))

No more defined symbols left to analyse.

(55) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n¹) was proven with the following lemma:
mark(gen_tt:s:0':U11:U12:plus2_0(n7802_0)) → gen_tt:s:0':U11:U12:plus2_0(n7802_0), rt ∈ Ω(1 + n7802₀)

(0) Obligation:

(1) DecreasingLoopProof (EQUIVALENT transformation)

(2) BOUNDS(n^1, INF)

(3) RenamingProof (EQUIVALENT transformation)

(4) Obligation:

(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

(6) Obligation:

(7) OrderProof (LOWER BOUND(ID) transformation)

(8) Obligation:

(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(10) Obligation:

(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

(12) Obligation:

(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(14) Complex Obligation (BEST)

(15) Obligation:

(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(17) Complex Obligation (BEST)

(18) Obligation:

(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(20) Complex Obligation (BEST)

(21) Obligation:

(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(23) Complex Obligation (BEST)

(24) Obligation:

(25) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(26) Complex Obligation (BEST)

(27) Obligation:

(28) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(29) Complex Obligation (BEST)

(30) Obligation:

(31) RewriteLemmaProof (LOWER BOUND(ID) transformation)

(32) Complex Obligation (BEST)

(33) Obligation:

(34) LowerBoundsProof (EQUIVALENT transformation)

(35) BOUNDS(n^2, INF)

(36) Obligation:

(37) LowerBoundsProof (EQUIVALENT transformation)

(38) BOUNDS(n^2, INF)

(39) Obligation:

(40) LowerBoundsProof (EQUIVALENT transformation)

(41) BOUNDS(n^2, INF)

(42) Obligation:

(43) LowerBoundsProof (EQUIVALENT transformation)

(44) BOUNDS(n^2, INF)

(45) Obligation:

(46) LowerBoundsProof (EQUIVALENT transformation)

(47) BOUNDS(n^2, INF)

(48) Obligation:

(49) LowerBoundsProof (EQUIVALENT transformation)

(50) BOUNDS(n^2, INF)

(51) Obligation:

(52) LowerBoundsProof (EQUIVALENT transformation)

(53) BOUNDS(n^2, INF)

(54) Obligation:

(55) LowerBoundsProof (EQUIVALENT transformation)

(56) BOUNDS(n^1, INF)